An upper bound for the first nonzero Neumann eigenvalue

Abstract

Let $\mathbb{M}$ denote a complete, simply connected Riemannian manifold with sectional curvature $K_{\mathbb{M}}\le k$ and Ricci curvature ${\text{Ric}}_{\mathbb{M}}\ge (n-1)K$, where $k,K\in \mathbb{R}$. Then for a bounded domain $\Omega \subset \mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue ${\mu }_1(\Omega )\le \mathcal{C}{\mu }_1(B_k(R))$. Here $B_k(R)$ is a geodesic ball of radius $R>0$ in the simply connected space form ${\mathbb{M}}_k$ such that vol$(\Omega )$ = vol$(B_k(R))$, and $\mathcal{C}$ is a constant which depends on the volume, diameter of $\Omega$ and the dimension of $\mathbb{M}$.